Local Lipschitz continuity of the stop operator

On a closed convex set Z in ^N with sufficiently smooth (W ^2,) boundary, the stop operator is locally Lipschitz continuous from W ^1,1([0,T]^N) × Z into W ^1,1([0,T],^N). The smoothness of the boundary is essential: A counterexample shows that C ¹-smoothness is not sufficient.     

On lengths,areas and Lipschitz continuity of polyharmonic mappings

In this paper, we continue our investigation of polyharmonic mappings in the complex plane. First, we establish two Landau type theorems. We also show a three circles type theorem and an area version of the Schwarz lemma. Finally, we study Lipschitz continuity of polyharmonic mappings with respect to the distance ratio metric.     

Lipschitz continuity theorems for free discontinuity problem in several variables

In this paper, following the method in [S. Solimini, Simplified excision techniques for free discontinuity problems in several variables, J. Funct. Anal. 151 (1997) 1-34], we prove a regularity of the function in minimizer for free discontinuity problem. Namely, we prove that the function is globally Lipschitz continuous out of a small neighborhood of the singular set.     

Lipschitz continuity ofV-subdifferentials of convex operators

It is proved that theV-subdifferential of a convex operator is locally Lipschitzian on the set of points at which it is continuous and subdifferentiable.Proposition 2.2 was originally stated for Holder continuity ofV-subdifferentials. The author would like to thank J. P. Penot for a very helpful suggestion which led to the present form of this proposition.     

Lipschitz continuity of the gradient of a one-parametric class of SOC merit functions

In this article, we show that a one-parametric class of SOC merit functions has a Lipschitz continuous gradient; and moreover, the Lipschitz constant is related to the parameter in this class of SOC merit functions. This fact will lay a building block when the merit function approach as well as the Newton-type method are employed for solving the second-order cone complementarity problem with this class of merit functions.     

Lipschitz continuity and duality for dynamic oligopolistic market equilibrium problem with memory term

We are concerned with an infinite dimensional variational inequality which is connected with the dynamic oligopolistic market equilibrium problem. We will provide existence theorems and show, under minimal assumptions on the data, the Lipschitz continuity of the solution. Moreover a general duality theory is provided overcoming the difficulty of the voidness of the interior of the ordering cone which defines the cone constraints.     

Lipschitz continuity of an approximate solution mapping for parametric set-valued vector equilibrium problems

The purpose of this paper is to generalize and improve some topological properties of solutions set to the set-valued vector equilibrium problems by using the scalar characterization method. Moreover, the Lipschitz continuity of an approximate solution mapping for the parametric set-valued vector equilibrium problems is studied.     

Lipschitz continuity of solutions of a free boundary problem involving the p-Laplacian

We prove local interior and boundary Lipschitz continuity of solutions of a free boundary problem involving the p-Laplacian.     

Lipschitz continuity of the optimal value function and KKT solution set in indefinite quadratic programs

When all the involved data in indefinite quadratic programs change simultaneously,we show the locally Lipschtiz continuity of the KKT set of the quadratic programming problem firstly, then we establish the locally Lipschtiz continuity of the KKT solution set. Finally, the similar conclusion for the corresponding optimal value function is obtained.     

CUTTING CORNERS PRESERVES LIPSCHITZ CONTINUITY

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Heinz‐type inequality and bi‐Lipschitz continuity for quasiconformal mappings satisfying inhomogeneous biharmonic equations

Let be the class of all sense‐preserving homeomorphic self‐mappings of . The aim of this paper is twofold. First, we obtain Heinz‐type inequality for (K,K′)‐quasiconformal mappings satisfying inhomogeneous biharmonic equation Δ(Δω) = g in unit disk with associated boundary value conditions and . Second, we establish biLipschitz continuity for (K,K′)‐quasiconformal mappings satisfying aforementioned inhomogeneous biharmonic equation when and are small enough.     

On the Lipschitz perturbation of monotonic functions

Summary A real valued function <InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"13"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"14"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"15"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"16"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"17"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"18"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"19"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"20"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"21"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"22"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"23"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"24"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"25"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"26"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"27"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"28"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"29"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"30"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"31"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"32"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"33"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"34"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>f$ defined on a real interval $I$ is called \emph{$d$-Lipschitz} if it satisfies $|\ell(x)- \ell(y)| \le d(x,y)$ for $x,y\in I$. In this paper, we investigate when a function $p\: I \to \bR$ can be decomposed in the form $p=q+ \ell$, where $q$ is increasing and $\ell$ is $d$-Lipschitz. In the general case when $d\: I^{2} \to \bR$ is an arbitrary semimetric, a function $p\: I \to \bR$ can be written in the form $p=q+ \ell$ if and only if \vspace{-4pt} <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[$$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> \sum_{i=1}^{n}{\big(p(s_{i})-p(t_{i})-d(t_{i},s_{i}) \big)^{+}} \le \sum_{j=1}^{m}{\big(p(v_{j})-p(u_{j})+d(u_{j},v_{j}) \big)} \vspace{-4pt} $$ is fulfilled for all real numbers $t_{1}<s_{1}, \dots, t_{n}<s_{n}$ and $u_{1}<v_{1}, \dots, u_{m}<v_{m}$ in $I$ satisfying the condition \vspace{-4pt} $$ \sum_{i=1}^{n} 1_{\left]t_i,s_i\right]}= \sum_{j=1}^{m} 1_{\left]u_j,v_j\right]}, \vspace{-4pt} $$ where $1_{\left]a,b\right]}$ denotes the characteristic function of the interval $\left]a,b\right]$. In the particular case when $d\: I^{2} \to R$ is a so-called concave semimetric, a function $p\: I \to \bR$ is of the form $p=q+ \ell$ if and only if \vspace{-4pt} $$ 0 \le \sum_{k=1}^{n}{d(x_{2k-1},x_{2k})} + d(x_0,x_{2n+1}) + \sum_{k=0}^{n}{\big(p(x_{2k+1})-p(x_{2k})\big)} \vspace{-4pt} $$ holds for all $x_0\le x_1\ki \cdots\ki x_{2n}\le x_{2n+1}$ in $I$.     

Lipschitz Continuity of inf-Projections

It is shown that local epi-sub-Lipschitz continuity of the function-valued mapping associated with a perturbed optimization problem yields the local Lipschitz continuity of the inf-projections (= marginal functions, = infimal functions). The use of the theorem is illustrated by considering perturbed nonlinear optimization problems with linear constraints.     

Lipschitz continuity of refinable functions on the Heisenberg group

In this paper, the Lipschitz continuity of refinable functions related to the general acceptable dilations on the Heisenberg group will be investigated in terms of the uniform joint spectral radius. We also give an investigation of the refinable functions in the generalized Lipschitz spaces related to a kind of special acceptable dilations.     

Lipschitz properties of the scalarization function and applications

The scalarization functions were used in vector optimization for a long period. Similar functions were introduced and used in economics under the name of shortage function or in mathematical finance under the name of (convex or coherent) measures of risk. The main aim of this article is to study Lipschitz continuity properties of such functions and to give some applications for deriving necessary optimality conditions for vector optimization problems using the Mordukhovich subdifferential.     

Preservation of Lipschitz constants by Bernstein type operators

Summary We obtain preservation inequalities for Lipschitz constants of higher order in simultaneous approximation processes by Bernstein type operators. From such inequalities we derive the preservation of the corresponding Lipschitz spaces.     

On Lipschitz behaviour of some generalized derivatives

The aim of the present paper is to compare various forms of stable properties of nonsmooth functions at some points. By stable property we mean the Lipschitz property of some generalized derivatives related only to the reference point. Namely we compare Lipschitz behaviour of lower Clarke derivative, lower Dini derivative and calmness of Clarke subdifferential. In this way, we continue our study of λ-stable functions.     

A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients

This paper is devoted to the analysis of measure-valued solutions to a nonlinear structured population model given in the form of a nonlocal first-order hyperbolic problem on R⁺. We show global existence and Lipschitz continuity with respect to the model ingredients. In distinction to previous studies, where the L¹ norm was used, we apply the flat metric, similar to the Wasserstein W₁ distance. We argue that analysis using this metric, in addition to mathematical advantages, is consistent with intuitive understanding of empirical data. Lipschitz continuous dependence with respect to the model coefficients and initial data and the uniqueness of the weak solutions are shown under the assumption on the Lipschitz continuity of the kinetic functions. The proof of this result is based on the duality formula and the Gronwall-type argument.